# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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<|e|2) = AkT0RnAf [V2] (2.7)

and perhaps one or more deterministic voltage generators. According to custom a noise voltage generator in a diagram is shown with the symbol of a normal voltage generator with Rn beside, as seen in Figure 2.2.

2.2.2 The equivalent noise conductance

Definition 2.2 The equivalent noise conductance of a one-port Gn is defined as

c. * Jgy И («)

where (|i|2) [A2] is the mean square (short circuited) noise current at

the terminals of the one-port in the frequency band Д/ [Hz], к = 1.3807 X Ш J K-1 is Boitzmann’s constant and To = 290 К is the standard noise temperature.

The same remarks as above for the eaui valent noise resistance can be added

here

conductance, admittance and parallel connection respectively. Similarly, a noise

'Tllis means that if Д/ ^ (1 +г)Д/ then it follows that ( |ej^) —- (1 -fe)( je|2} when £ is a small number.

2.2. Definitions of noise quantities

13

current generator is shown with the symbol of a normal current generator with a Gn beside and the value ol’ the current is determined by

(ii|2) = 4 к TaGr. А/ [A'2] (2.9)

2.2.3 The extended noise temperature

The above definitions for Rn and Gn have been chosen in such a way that they are both positive whatever sign the immittance of the one-port has. In the following definition of the noise temperature for a one-port the usual definition [4] has been extended such that the sign of the extended noise temperature shows if the one-port is active8 or passive [61.

Definition 2.3 The extended noise temperature of a one-port Tem is the exchangeable noise power density, N, [W/Ilzj, divided by Boltzmann’s constant, к — 1.3807 x 10_2J J K-1.

Tcm = ^ [K] (2.10)

This definition is equivalent to that in Equations (2.4) and (2.5) keeping the values of R and G at their physical values and then adjusting T until the equations are fulfilled. Thus definition 2.3 can be expressed by either

Г - < H2)

4 kRAf

or

т - (i‘l2)

4 kGAf

The subscript em stands for extended, which refers to the extension of the noise temperature to active one-ports, and mono for one-port.

The extended noise temperature is negative when the one-port is active as Y' then is negative. This corresponds to a negative R or G in Equations (2.11) and (2.12). Of course T.m can not tell anything about the physical temperature except in the case of pure thermal noise (ot a one-port which then must be passive) where 7',,,, equals the physical temperature From

(H2) = 4 к Тц Rn Af = 4kTemR±f

(hi2) = ikToG^Af = -IkT^GAf

the following nsnfui relation? яге easily derived:

Trm _ Rn _ Gn

To R Ci

[K]

[K]

(2.11)

(2.12)

’Active means a one port with negative teal part of its immittance.

2.3 Calculation with noise quantities

In order to calculate the noise properties of series and parallel connections of uncorrelated9 one-ports it is necessary to develop rules for series connection of one-ports characterized by either Rn or Tcm and similarly parallel connection of one-ports characterized by either Gn or Tem.

Noise generators are represented by stochastic processes, and when those are uncorrelated (independent) the following formulas for the equivalent mean square noise voltage ( |e|2) for series connected aoise voltage generators and the equivalent mean square noise current ( U'j2) for parallel connected noise current generators are valid:

(И2) = Y,J2(eie') = (Ы2> + <N2} + + (lp-/!2> [v2] (2.i3)

1=1 J=1

(l‘f> = EX>*‘) = (I'll2) + (N2> + ••• + <H2> [A2] (2.14)

■=ij=l

2.3.1 Series and parallel connections of one-ports characterized by equivalent noise resistances and conductances

The Thevenin equivalent of a noisy one-port is a series connection of an internal impedance Z (passive or active), a (stochastic) noise voltage generator Rn generating a mean square noise voltage given by Equation (2.4) and perhaps a (deterministic) voltage generator E аз shown in Figure '2.3.

From circuit theory the equivalent one-port of I series connected one-ports as shown in Figure 2.3 has Z = ]Tf=i Zt [fi] and E = J2i=i t^l- As the noise contributions from the different one-ports arc uncorrelated Equations (2.4) and (2.13) give

/

4 к To Rn А/ = £4 kT0Rn,,Af fV2]

J = 1

which leans to

1

ii„ - Hn i [wj (2.15}

9 Physically separated one-ports generally have stochastic independent and thus unconelated noise generators.

2.3. Calculation with noise quantities

15

Figure 2.3: Series connection of noisy one-ports and the corresponding equivalent circuit.

In a very similar way, from Equations (2.5) and (2.14) it is found that

Gn

= XX

(2.16)

where I is the number of parallel connected one-ports.

Example 2.1 In the figure a one-port is shown where G\, G-i and Д3 are known. Also known are G2'5 equivalent noise conductance (S'„j and R3's equivalent noise resistance Rn 3. When the one-port as such Has the noise temperature Tem, Gi’s equivalent noise conductance G>.,i can be

Gi

Gn.i

13 rtn.3

G: Gn, 2

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